Resource Allocation Over Time

A complete guide to nonrenewable-resource use across generations

A study guide to marginal net benefit, present value, user cost, depletion taxes, scarcity rent, Hotelling’s rule, and the Hartwick rule.

NoteFocus questions

1. How should a limited nonrenewable resource be divided across generations?
2. How can benefits and costs at different dates be compared?
3. How do scarcity, prices, and consumption change as a resource is depleted?

1. Renewable and nonrenewable resources

A renewable resource can regenerate through ecological processes. Farms, forests, and fisheries can remain productive for long periods when harvest does not continually exceed renewal. Renewable does not mean impossible to deplete.

A nonrenewable resource does not regenerate on a human time scale. Oil, coal, and mineral deposits are common examples. Extracting a unit today reduces the physical stock available later.

The economic question is not simply whether a nonrenewable stock will decline. The question is how to compare the value of using one more unit now with the value of preserving it for future use. Technology, substitution, new discoveries, and recycling can change effective scarcity, but they do not remove the need to make choices across time.

This guide studies recoverable neodymium in a known deposit using a two-period model. The model is deliberately simple. It reveals the main logic that carries into models with many periods.

2. The neodymium market in one period

Neodymium in this model

Illustration of a neodymium supply chain from a mineral deposit through separated material to a permanent-magnet motor.

Neodymium is a rare earth element used in high-strength permanent magnets. These magnets appear in electric motors and some wind-turbine generators. Rare earth elements are relatively abundant in Earth’s crust, but mineable concentrations are less common.

The model tracks 250 hypothetical tons of recoverable neodymium in one known deposit. Quantity measures recoverable neodymium content rather than the mass of raw ore. The supply curve includes marginal mining, separation, and processing cost.

Actual ores contain several rare earth elements that producers recover jointly. The model temporarily treats neodymium as a single product. Its equations and numerical values illustrate the economic logic rather than estimate the current neodymium market.

Supply, demand, and the static equilibrium

Suppose this illustrative neodymium market has these inverse demand and supply equations:

P_D=150-0.25Q,

P_S=50+0.25Q.

Demand measures permanent-magnet manufacturers’ marginal willingness to pay. Supply measures marginal mining, separation, and processing cost.

Setting the curves equal gives the one-period equilibrium:

150-0.25Q=50+0.25Q,

Q=200,\qquad P=100.

Figure 1: If only current benefits and costs matter, the illustrative neodymium market clears at 200 model tons and $100 per ton.

This is a static equilibrium because the calculation counts only benefits and costs in the current period.

Marginal net benefit

The marginal net benefit (MNB) of one more ton is its marginal benefit minus its marginal cost. It compresses the demand and supply information into one curve:

MNB=P_D-P_S.

For this market,

MNB=(150-0.25Q)-(50+0.25Q)=100-0.5Q.

At low quantities, buyers’ willingness to pay substantially exceeds extraction cost. MNB falls as quantity increases. It reaches zero at Q=200, where the market reaches its static equilibrium. Production beyond 200 would have negative marginal net benefit.

Figure 2: The area under MNB through 200 model tons is the total net benefit of the illustrative neodymium market.

The area under MNB is total net benefit, which equals total benefit minus total cost. Here it is the area of a triangle:

TNB=\frac12(200)(100)=10{,}000.

Total net benefit reaches its maximum when MNB reaches zero. This reproduces the one-period market equilibrium.

3. A fixed stock across two periods

Suppose 250 model tons of recoverable neodymium must be divided between Period 1 and Period 2:

Q_1+Q_2=250.

This supply constraint means that additional use in one period leaves exactly that much less for the other period. Assume that the demand, supply, and MNB functions are initially the same in both periods:

MNB_1=100-0.5Q_1,

MNB_2=100-0.5Q_2.

The mirror-axis graph reads Q_1 from left to right and Q_2 from right to left. Every point on the horizontal axis allocates all 250 tons.

Figure 3: The mirror axis keeps the 250-ton supply constraint visible: moving right increases Period 1 use and reduces Period 2 use by the same amount.

4. Present value and discounting

Present value converts a future benefit or cost into its value at a common earlier date. With annual discount rate r and n years between dates,

PV(X_n)=\frac{X_n}{(1+r)^n}.

The example uses a 7.25% annual rate and periods ten years apart. Five hundred dollars invested today at roughly 7.25% grows to about $1,000 after ten years. Equivalently, the present value of $1,000 received with certainty in ten years is about $500:

\frac{1{,}000}{(1.0725)^{10}}\approx 500.

The example assumes a real interest rate, meaning that expected inflation has already been removed. A person who prefers current cash could borrow against the certain future payment. That borrowing interpretation explains why two dated amounts can be economically equivalent.

For the neodymium example, the ten-year discount factor is approximately two:

PV[MNB_2]=\frac{MNB_2}{(1.0725)^{10}}\approx\frac{MNB_2}{2}.

Discounting therefore halves every Period 2 marginal net benefit when measured in Period 1 dollars.

5. Dynamic equilibrium

The efficient allocation makes Period 1 MNB equal to the present value of Period 2 MNB:

MNB_1=PV[MNB_2].

Together with the stock constraint, the model has two equations:

100-0.5Q_1=\frac{100-0.5Q_2}{2},

Q_1+Q_2=250.

Substituting Q_2=250-Q_1 gives

100-0.5Q_1=50-0.25(250-Q_1),

0.75Q_1=112.5,

Q_1=150,\qquad Q_2=100.

Figure 4: Figure 5.4 concept. At 150 tons now and 100 tons later, the marginal net benefits are equal after discounting, and combined discounted net benefit is maximized.

A dynamic equilibrium counts both present and future benefits and costs. The crossing maximizes the sum of Period 1 net benefit and the present value of Period 2 net benefit.

Why another allocation loses welfare

Consider Q_1=200 and Q_2=50. Shifting 50 tons from Period 2 to Period 1 adds some current net benefit, but the discounted future benefit lost is larger. The difference is a welfare loss.

Figure 5: Figure 5.5 concept. Moving from the efficient allocation to 200 tons in Period 1 sacrifices more discounted future benefit than it adds in current benefit.

The opposite shift, such as Q_1=100 and Q_2=150, also moves away from the crossing and lowers discounted total net benefit.

6. User cost

Using a resource today can reduce the benefit available to future users. The lost future opportunity is the user cost of current extraction.

In this example, Period 2 would use at most 200 tons in its static market. If Period 1 uses no more than 50 tons, at least 200 remain, so one more current ton does not yet reduce the desired Period 2 quantity. Beyond 50 tons today, each additional current ton displaces a future ton. Its present-value loss appears as the rising PV[MNB_2] curve.

At the efficient allocation,

\text{User cost}=MNB_1(150)=100-0.5(150)=25,

and the same value appears from the future side:

\text{User cost}=PV[MNB_2(100)]=\frac{100-0.5(100)}{2}=25.

This framework describes user cost as an externality across time when the current market leaves the future loss out of current decisions. It also explains that forward-looking owners may anticipate scarcity and incorporate some or all of that value themselves.

Evidence from Jordan

ImportantEstimating user costs in Jordan

Jordan exports phosphate and potash used in fertilizer. A study of extraction from 2002 through 2010 estimated user costs of about $550 million using a 3% discount rate, roughly 20% of mining profits during the period.

The central sustainability issue was what happened to those proceeds. Using depletion income mainly for current consumption would leave fewer assets for future generations. Reinvesting the user-cost portion in education and skills, produced capital, or renewable natural capital could preserve a future income stream. The case therefore links measurement of user cost with governance and reinvestment.

The source note in the textbook attributes the estimate to Alrawashdeh and Al-Tarawneh (2014). The surrounding discussion gives a different publication year, so the primary source should be checked before citing its year independently.

7. Resource depletion policy

Social cost and the depletion tax

If the current market ignores user cost, it chooses 200 tons. Adding marginal user cost to extraction cost creates a social-cost schedule. A resource depletion tax can reproduce the efficient outcome when it equals the marginal user cost at the target quantity.

At Q_1=150, the correct tax is $25 per ton. The tax shifts the buyer-facing supply curve to

P=75+0.25Q.

Equating it with demand gives

150-0.25Q=75+0.25Q,

Q_1=150,\qquad P_1=112.50.

Figure 6: Figure 5.6 concept. User cost raises the full social cost of current extraction. A $25 depletion tax reaches the efficient Period 1 quantity of 150 tons.

The social-cost curve and the constant tax-adjusted supply curve differ away from the efficient quantity. They coincide at the efficient quantity because the tax equals marginal user cost there.

The second-period outcome

With 150 tons used in Period 1, 100 remain for Period 2. The demand curve gives

P_2=150-0.25(100)=125.

Figure 7: Figure 5.7 concept. The remaining stock fixes second-period supply at 100 tons, and demand determines a price of $125.

The higher current price discourages some present use and preserves more neodymium for the future. Other possible policies include direct limits on extraction, setting deposits aside, or maintaining public stockpiles of processed material.

8. Scarcity rent and private owners

Government intervention may be unnecessary when private owners can foresee scarcity, control extraction, and capture the future value of leaving resources underground.

To see the incentive, consider the static allocation of 200 tons now and 50 tons later. At 50 tons in Period 2,

P_D(50)=137.50,

P_S(50)=62.50.

The $75 difference is scarcity rent on the marginal unit:

137.50-62.50=75.

Scarcity rent is the payment above the amount needed to cover the marginal production cost. A forward-looking owner may withhold some current supply to earn larger discounted rent later. Under restrictive assumptions, profit maximization can produce the same two-period allocation as the depletion tax by equalizing discounted scarcity rents.

Reasons for caution remain. Owners may have short horizons, uncertain information, insecure claims, limited access to finance, or incentives that differ from social objectives. The relevant question is whether current market prices already contain the future scarcity value.

9. Discount rates change the allocation

With a zero discount rate, the two periods receive equal weight and the stock divides evenly: 125 tons in each period. A positive discount rate gives more weight to current benefits. As the rate rises, the efficient allocation shifts toward Period 1 and user cost falls.

Figure 8: Figure 5.8 concept. Larger discount factors rotate the present-value curve downward, moving the crossing toward greater Period 1 use.

Annual discount rate	Ten-year factor, (1+r)^{10}	Q_1	Q_2
0%	1.0	125	125
2%	1.2	132	118
5%	1.6	143	107
7.5%	2.0	150	100
10%	2.6	158	92
15%	4.0	170	80
20%	6.2	179	71
30%	13.8	190	60

At 30%, the allocation approaches the static quantity of 200 tons in Period 1 and the user cost becomes small. Discounting over very long horizons can make distant benefits carry little present weight.

10. Hotelling’s rule

The two-period logic extends to many dates. Hotelling’s rule states that, in equilibrium, the net price of a nonrenewable resource rises at the interest rate:

R_{t+1}=(1+r)R_t,

where net price or resource rent is market price minus marginal extraction cost.

The owner’s choice explains the rule:

• If current rent invested at interest would exceed expected future rent, extracting now pays more.
• If expected future rent would exceed current rent plus interest, waiting pays more.
• Extraction adjusts until the owner is indifferent at the margin.

With an initial net price of 100 and a 7% rate,

R_t=100(1.07)^t.

Figure 9: Figure 5.9 concept. In the benchmark, net price follows an exponential path whose growth rate equals the interest rate.

Higher interest rates encourage faster extraction because earning a market return on current proceeds becomes more attractive. Economic theory therefore implies an optimal depletion rate that maximizes the resource’s net present value. Under the benchmark assumptions, complete exhaustion can be part of that optimum.

Observed resource prices need not follow a smooth Hotelling path. New discoveries, technical change, changing extraction costs, market power, taxes, uncertainty, environmental policy, substitution, and recycling can all alter the observed price.

11. The Hartwick rule and future generations

Complete physical depletion raises an ethical question. The Hartwick rule distinguishes conserving a particular deposit from preserving productive capacity for future people. It says that society should invest scarcity rents from nonrenewable extraction rather than consume them:

\text{scarcity rent}=\text{resource revenue}-\text{extraction cost}.

NoteNatural capital

Natural capital is the stock of natural resources and ecosystems that generates benefits over time. For example, a standing forest is natural capital because it can provide timber while also storing carbon, regulating water flows, and supporting wildlife habitat.

Investment in skills, infrastructure, technology, or renewable natural capital can leave future generations other productive assets. This is a weak-sustainability argument because it permits produced capital to replace depleted natural capital.

A key limitation is that produced assets may not adequately replace natural capital with unique ecological, cultural, or life-support functions. The substitution argument is more plausible for a mineral deposit with little separate ecological value than for a critical ecosystem.

12. Limits and extensions

Discounting and distant generations

A market interest rate can place very little weight on effects far in the future. That raises a normative question: should commercial returns determine how society values the welfare of distant generations? Cost-benefit analysis often distinguishes a social discount rate from private market returns because the choice contains ethical as well as financial judgments.

Environmental externalities

The basic neodymium model assumes that the supply curve captures all social extraction costs. Mining, ore concentration, and chemical separation can affect water, landscapes, habitat, and human health while generating tailings and other wastes. Adding those costs would change the efficient price and extraction path.

Recycling and additional supply

The model also omits recycling. Recovering neodymium from manufacturing scrap or end-of-life permanent magnets can add secondary supply and reduce pressure on newly mined deposits. New discoveries and technical change can similarly expand economically recoverable supply. These extensions alter the numerical path without removing the intertemporal logic.

Assumptions behind the model

• Rare earth elements are relatively abundant in Earth’s crust, but mineable concentrations are less common.
• The 250-ton stock represents hypothetical recoverable neodymium content rather than raw ore mass or a measured real-world reserve.
• The model initially ignores joint production with other rare earth elements, recycling, substitution, and new discoveries.
• Periods are ten years apart, and all remaining stock is used by Period 2.
• Supply and demand functions are unchanged between periods.
• The 7.25% rate is real rather than nominal.
• The benchmark initially omits environmental externalities.
• Hotelling’s rule originates with Harold Hotelling’s 1931 analysis and has mixed empirical performance for observed commodity prices.

Further reading and source notes

• Later portions of the source text develop recycling, add extraction externalities, and examine evidence on resource prices and extraction.
• The investment rule is traced to Hartwick (1977) and Solow (1986).

13. Summary

1. A nonrenewable stock used today cannot also be used later. Efficient allocation compares current marginal net benefit with the present value of the future marginal net benefit forgone.
2. Present value places dated benefits and costs on one clock. A higher discount rate shifts more use toward the present.
3. The illustrative one-period neodymium market selects 200 model tons. With a 250-ton stock and a ten-year discount factor of two, the dynamic allocation is 150 tons now and 100 tons later.
4. User cost measures the lost future opportunity created by current extraction. At the efficient Period 1 quantity, it is $25 per ton.
5. A $25 resource depletion tax can internalize this user cost, raising the Period 1 price to $112.50 and reducing current use to 150 tons.
6. Forward-looking owners may capitalize scarcity into prices. Scarcity rent gives them an incentive to withhold some current supply.
7. Hotelling’s rule says the net resource price rises at the interest rate in the benchmark model. Higher rates imply faster depletion.
8. The Hartwick rule directs scarcity rents into investment for future well-being, but its adequacy depends on whether other capital can substitute for the depleted natural asset.
9. Long-horizon discounting, environmental damage, uncertainty, market imperfections, recycling, substitution, discovery, and technical change limit the simple model.

14. Key terms

Term	Meaning here
Renewable resource	A resource regenerated through ecological processes, although excessive use can still deplete it
Nonrenewable resource	A resource that does not regenerate on a human time scale
Recoverable neodymium	Neodymium content in a known deposit that can be produced under the model’s assumed technology and costs
Marginal net benefit	Marginal benefit minus marginal cost for one more unit
Total net benefit	Total benefit minus total cost, represented by the area under MNB
Static equilibrium	An equilibrium based only on current benefits and costs
Present value	A future value expressed in current-value terms
Discount rate	The rate used to translate future benefits and costs into present values
Supply constraint	The fixed total stock that must be allocated across dates
Dynamic equilibrium	An equilibrium that counts present and future benefits and costs
User cost	The opportunity cost of future use lost through current extraction
Social cost	Market and nonmarket costs associated with a good or service
Resource depletion tax	A tax on extraction or sale designed to reflect user cost
Scarcity rent	The resource price net of the marginal cost needed to supply it
Hotelling’s rule	The benchmark condition that net resource price rises at the interest rate
Optimal depletion rate	The extraction path that maximizes the resource’s net present value
Hartwick rule	The principle that scarcity rents should be invested rather than consumed

15. Discussion questions

1. Private traders may anticipate scarcity and conserve a resource for future profit. Under what conditions is that argument convincing? When might public policy improve the outcome, and which policy tool would fit the problem?
2. Can this framework be applied to the atmosphere or oceans? Identify where a fixed-stock model helps and where regeneration, common access, ecosystem thresholds, or pollution make its conclusions incomplete.

References

• Harris, Jonathan M., and Brian Roach. 2021. Environmental and Natural Resource Economics: A Contemporary Approach, 5th ed. “Resource Allocation Over Time,” pp. 119–135.
• Hartwick, John M. 1977. “Intergenerational Equity and the Investing of Rents from Exhaustible Resources.” American Economic Review 67 (5): 972–974. JSTOR.
• Solow, Robert M. 1986. “On the Intergenerational Allocation of Natural Resources.” The Scandinavian Journal of Economics 88 (1): 141–149. https://doi.org/10.2307/3440280.
• U.S. Department of Energy. 2022. Rare Earth Permanent Magnets: Supply Chain Deep Dive Assessment. Report.
• U.S. Geological Survey. 2026. “Rare Earths.” Mineral Commodity Summaries 2026. https://doi.org/10.3133/mcs2026.